cp's OEIS Frontend

A083058 Number of eigenvalues equal to 1 of n X n matrix A(i,j)=1 if j=1 or i divides j.

%I A083058 #53 Jan 14 2024 02:33:11
%S A083058 1,0,1,1,2,3,4,4,5,6,7,8,9,10,11,11,12,13,14,15,16,17,18,19,20,21,22,
%T A083058 23,24,25,26,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,
%U A083058 45,46,47,48,49,50,51,52,53,54,55,56,57,57,58,59,60,61,62,63,64,65,66
%N A083058 Number of eigenvalues equal to 1 of n X n matrix A(i,j)=1 if j=1 or i divides j.
%C A083058 All numbers occur at least once, but terms > 1 of A000295 appear twice. - _Robert G. Wilson v_, Apr 19 2006
%C A083058 It appears that a(n) = Sum_{k=0..n-1} (1 + (-1)^A000108(k))/2 (n > 1). - _Paul Barry_, Mar 31 2008
%C A083058 Barry's observation above is true because A000108 obtains odd values only at points (2^j)-1 (A000225) and here the repeated values (A000295) occur precisely at positions given by A000225 and A000079. - _Antti Karttunen_, Aug 17 2013
%C A083058 a(n)+1 gives a lower bound for nonzero terms of A228086 and A228087. - _Antti Karttunen_, Aug 17 2013
%H A083058 Antti Karttunen, <a href="/A083058/b083058.txt">Table of n, a(n) for n = 1..8192</a>
%H A083058 J. B. Conrey, <a href="http://www.ams.org/notices/200303/fea-conrey-web.pdf">The Riemann Hypothesis</a>, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347.
%H A083058 Will Dana, <a href="https://sites.math.washington.edu/~morrow/336_15/papers/will.pdf">Eigenvalues of the Redheffer Matrix and their relation to the Mertens function</a> (2015), Theorem 5.
%H A083058 Ralf Stephan, <a href="/somedcgf.html">Some divide-and-conquer sequences with (relatively) simple ordinary generating functions</a>.
%H A083058 Ralf Stephan, <a href="/A079944/a079944.ps">Table of generating functions</a>.
%F A083058 a(n) = n - A070939(n), n > 1.
%F A083058 a(1)=1, else a(n)=b(n) with b(0)=0, b(2n)=b(n)+n-1, b(2n+1)=b(n)+n. - _Ralf Stephan_, Oct 11 2003
%F A083058 Except for a(1), a(n) = n - 1 - floor(log(2,n)). - _Robert G. Wilson v_, Apr 19 2006
%F A083058 It seems that a(n) = A182220(n+1)-1 for all n > 1. - _Antti Karttunen_, Aug 17 2013
%p A083058 A083058 := proc(n)
%p A083058     if n = 1 then
%p A083058         1;
%p A083058     else
%p A083058         n-floor(log[2](n))-1 ;
%p A083058     end if;
%p A083058 end proc:
%p A083058 seq(A083058(n),n=1..40) ; # _R. J. Mathar_, Jul 23 2017
%t A083058 a[1] = 1; a[n_] := n - Floor[Log[2, n]] - 1;
%t A083058 Array[a, 100] (* _Jean-François Alcover_, Feb 27 2019 *)
%o A083058 (PARI) a(n)=if(n<2,n>0,n-floor(log(n)/log(2))-1)
%o A083058 (PARI) a(n)= if(n<1, 0, valuation( subst( charpoly( matrix(n, n, i, j, (j==1) || (0==j%i))), x, x+1), x))
%o A083058 (Scheme) (define (A083058 n) (if (< n 2) n (- n (A070939 n)))) ;; _Antti Karttunen_, Aug 17 2013
%o A083058 (Python) def a(n): return n - n.bit_length() + (n == 1)  # _Matthew Andres Moreno_, Jan 04 2024
%Y A083058 Cf. A002321, A070939, A143104.
%K A083058 nonn
%O A083058 1,5
%A A083058 _Michael Somos_, Apr 18 2003